# Statistics Resources

This guide contains all of the ASC's statistics resources. If you do not see a topic, suggest it through the suggestion box on the Statistics home page.

## Measures of Central Tendency

Measures of central tendency refer to the idea that we can capture a single representative value for a set of data points. As the name implies, these measures describe the "center" of the data, but each measure has a different idea of what "center" looks like. There are three main measures of central tendency:

• Mean
• Median
• Mode

The mean refers to the mathematical average of a set of numbers. The mean is computed by adding up all of the values in the dataset and dividing by the number of values that were added. Consider the following sample set:

12      17      21      13      11      15      17      18      20      17      11

Step 1: Add up the values in the set:

12 + 17 + 21 + 13 + 11 + 15 + 17 + 18 + 20 + 17 + 11 = 172

Step 2: Divide that sum by the number of values in the set:

172/11 = 15.6363

Thus, the average value in the set is 15.6 (rounded to one decimal place).

The mean is the best measure of central tendency if the data is:

• numerical (interval or ratio level of measurement)
• approximately normally distributed

The mean is not appropriate if the data:

• is categorical (nominal or ordinal level of measurement)
• is skewed
• has outliers

The median is a measure of the physical middle of the data. To find the median value, we place all of the values in order from least to greatest and then locate the value that divides the list in half. Consider the following sample set:

12      17      21      13      11      15      17      18      20      17

Step 1: Place the values in order from least to greatest.

11      12      13      15      17      17      17      18      20      21

Step 2: Identify the point that divides the data in half.

11      12      13      15      17   |   17      17      18      20      21

There are exactly five values below the middle and five values above the middle. Since the middle point is not on a number, we will average the points on either side of the line. To average the points, we add them together and divide by two: 17 + 17 = 34 / 2 = 17. The median of the sample set is 17.

If a data set has an odd number of values, the median will be the value that divides the list in half. Consider this sample set that has already been put in order:

11      11      12      13      15      17      17      17      18      20      21

The highlighted number has five values below it and five values above it, thus evenly dividing the list in half. The median of this sample set is 17.

The median is the best measure of central tendency if the data:

• is numerical, and its distribution is skewed
• is numerical and has outliers
• is ordinal

The mode represents the value that occurs the most in a set of values. You could also think of this value as having the highest frequency. A simple way of identifying the mode is to put all of the values in order from least to greatest and identify the value that repeats the most. Consider the following sample set:

12      17      21      13      11      15      17      18      20      17      11

Step 1: Put the list in order from least to greatest.

11      11      12      13      15      17      17      17      18      20      21

Step 2: Identify the value that repeats the most.

11      11      12      13      15      17      17      17      18      20      21

Since 17 repeats three times in this list, 17 is the mode of the data.

The mode is the best measure of central tendency if the data is categorical (nominal or ordinal level of measurement).

The mode is generally not ideal for numerical data (interval or ratio level of measurement).